By definition, it suffices to show \(f_{\mathcal {L},\zeta }=\tau _{\mathcal {L},\zeta }^*\)  for a path \(\zeta \in \Omega ^*\)  ending at a vertex in \(\mathcal {L}\) .
There are many ways to prove this result.
One could use a weighted version of the arguments in [1]}, which are proven in a similar irreducible matrix-product setting.
Another option is to follow the arguments in [2]}.
